3.2Lesson 3.2 · Module 3

Bond prices are not magic, but they talk.

A zero-coupon bond price tells us how the market values one future dollar. A set of zero-coupon prices becomes a yield curve. A yield curve lets us infer spot rates, forward rates, and the market's view of future interest rates — useful, but not a crystal ball.

Prices → Spot rates → Yield curve → Forward rates → Decisions
Decode notation before using it: $R_{t}$ vs $r_{0, T}$ vs $f_{t}$
Lock in future lending or borrowing with forward rates
Value coupon bonds as portfolios of zero-coupon bonds

Start Spot Rates, Forwards, and Coupon Bonds View module map

Lecture hook

In the lecture, Professor Lo begins with crisis-era market news. Markets had priced in a Fed rate cut — but the Fed did something different. That is the first lesson of this entire module: market prices contain information, but they are not perfect predictions. An $85 billion loan is "a lot of money," but the point is not the number. The point is that prices reflected one expectation, and reality did something else.

Learning objectives

By the end of this lesson, you should be able to:

1Explain what information a zero-coupon bond price contains.
2Define today's T-year spot rate and distinguish it from future one-year rates.
3Explain why a T-year spot rate is a geometric average representation of one-year rates.
4Infer spot rates from STRIPS prices.
5Compute a one-year forward rate from spot rates.
6Explain why a CFO might lock in a forward rate rather than speculate.
7Explain YTM as a single-rate summary of coupon-bond cash flows.
8Explain why coupon bonds can be valued as portfolios of pure discount bonds.

M3Module map

Fixed-Income Securities

Bonds move the financial system. Two lessons build the valuation machinery now; duration, convexity, credit risk, and securitization arrive in upcoming lessons.

Node 01

Bond Markets & Discount Bonds

The market system, cash flows, and zero-coupon valuation

Ready

Node 02

Spot Rates & Yield Curves

Forwards, YTM, replication, and arbitrage

Ready

Node 03

Arbitrage, Duration & Convexity

Law of one price, mispricing signals, and interest-rate risk

Ready

Node 04

Credit Risk & Securitization

Corporate bonds, default risk, spreads, and structured credit

Ready

01Chapter 1

Prices, spot rates, and the term structure

Before we introduce any notation, let's ask a simple question: what is a bond price actually telling us? The answer turns out to be surprisingly deep. A zero-coupon bond price tells us how the market values one specific dollar at one specific future date. And once we can read those prices, we can back out the interest rates the market is using.

1.1Section 1

Market prices are thermometers, not crystal balls

Market prices are like financial thermometers. They show what investors are willing to pay right now, given fear, liquidity, expectations, and available alternatives. But a thermometer is not a crystal ball. It gives a current reading, not a guaranteed future. Market prices implied one Fed outcome; the Fed did something else. The lesson is not that prices are useless — it is that prices contain information, but they can still be wrong.

Prices → Spot rates → Forward rates → Market-implied expectations.

What the market is not

A crystal ball

Bond prices do not know the future. They cannot guarantee what rates will be.

What the market is

A market thermometer

Prices show what investors are willing to pay today for dollars arriving at different future dates.

Definition · Reading the market

Market prices contain information, but they are not a perfect crystal ball. Bond prices can imply what the market is pricing about future interest rates, but market-implied information can be wrong.

Try itPrices → rates → curve

From scattered prices to a structured curve

Discount bond prices are floating in the market. Press “Read the market” to organize them into spot rates and connect them into a yield curve.

1.2Section 2

Start from the old problem: a pure discount bond

Last lesson, we priced pure discount bonds — bonds that pay only principal at maturity and no intermediate coupons. A pure discount bond is the cleanest instrument for learning interest rates because it has exactly one payment at exactly one date. No coupons to complicate things. That simplicity is what lets us extract a rate.

Definition · Why start with a pure discount bond?

A pure discount bond is the cleanest instrument for learning rates because it has one payment and one date. No coupons, no reinvestment ambiguity — just a price today and a face value at maturity.

Try itRecap from Lesson 3.1

One payment, one date, one rate

Today · t = 0

P₀

price today

Maturity · t = T

face value at maturity

If you know F, P₀, and T, you can solve for r. That single rate — the price-implied rate for the whole interval — is exactly what we will turn into the spot rate in this lesson.

Pure discount bond

P_{0} = \frac{F}{( 1 + r ) ^{T}}

$P_{0}$: price today
$F$: face value at maturity
$r$: discount rate
$T$: maturity in years

If you know F, P₀, and T, you can solve for r.

1.3Section 3

Why different horizons need different rates

A one-year rate is not necessarily the same as a five-year rate. The market can have different expectations about the economy, inflation, liquidity, and borrowing conditions at different horizons. So instead of asking "what is the interest rate?" — a question that has no single answer — ask: "What price would the market pay today for $1,000 in one year, $1,000 in two years, $1,000 in five years, and so on?" The market prices those pieces of paper. Once we have price and face value, we solve for the rate.

Definition · The price comes first

Markets do not announce rates. Markets announce prices. The rate is backed out from the price an investor is willing to pay for a future dollar.

Try itAuction · four claims on $1

Buy a dollar in the future. Read the rate off the price.

Each bond below pays exactly $1 at maturity and nothing before. Click a bond to discover what market price that implies for today's spot rate.

The price comes first. The rate is backed out from the price. Notice the longer you wait for your dollar, the lower the price — and the higher the implied rate.

r_{0, T} = (\frac{1}{P _{0}})^{1/ T} - 1

1.4Section 4

What is a zero-coupon bond, and what is a 5-year zero?

Definition · Zero-coupon bond

A zero-coupon bond makes no intermediate coupon payments. It pays one amount at maturity. A "5-year zero" means a zero-coupon bond that pays at year 5.

The 5-year zero example

0.797 = \frac{1}{( 1 + r _{0, 5} ) ^{5}}

Today you pay $0.797. In five years you receive $1. What annualized rate connects those two values?

r_{0, 5} = (\frac{1}{0.797})^{1/5} - 1

$r_{0, 5}$ ≈ 4.64%

The price 0.797 means that $1 delivered in five years is worth $0.797 today. The annualized rate connecting those values is today's 5-year spot rate.

Try itPrice → rate translator

Translate a price into today's spot rate

Price today (P₀, per $1 face)$0.797

Maturity (years)5 yr

Implied 5-year spot rate

4.64%

This price implies today's 5-year spot rate of 4.64%.

Solve the spot rate

r_{0, T} = (\frac{1}{P _{0}})^{1/ T} - 1

r_{0, 5} = (\frac{1}{0.797})^{1/5} - 1

r(0,5) ≈ 4.64%

A higher price means investors accept a lower rate to wait for that future dollar.

1.5Section 5

Decode spot-rate notation before using it

A spot rate is the rate observed today for money moving from today to a future date. Before we use the notation, let's decode it. The symbol $r_{0, 5}$ has two subscripts. The first subscript is the pricing date — today, time 0. The second subscript is the maturity date — year 5. Read it as: "the annualized rate, observed today, for money paid at year 5."

Definition · Spot rate notation

r_{0, T}

reads as r-sub-zero-T. The first subscript is the pricing date. The second subscript is the maturity date.

Try itNotation decoder

What do the subscripts mean?

r_{0, 5}

→today's 5-year spot rate

Pricing date

0 (today)

Maturity date

t = 5

First subscript = pricing date. Second subscript = maturity date. Both dates are measured from today.

1.6Section 6

Capital R versus lowercase r

This is the biggest conceptual hurdle in the entire lesson, so let's slow down. Professor Lo uses capital $R$ and lowercase $r$ to separate two different ideas:

Capital R

R_{t}

= a one-year rate for one specific one-year period.

R_{1}

= year 0→1.

R_{2}

= year 1→2. Always one year.

Lowercase r

r_{0, T}

= one annualized rate observed today for the whole interval from 0 to T.

Multi-year. Inferred from today's bond price.

Here is the key: we do not observe the entire future sequence of R's today. We do not know what $R_{2}$ or $R_{3}$ will actually be. We observe prices today, and from those prices we infer $r_{0, T}$ . The observable little r contains information about the market's view of the future path of rates — at least the market's current expectation of them.

Definition · Two different rate objects

Capital

R_{t}

is a one-year rate for one slice of time. Lowercase

r_{0, T}

is one annualized rate for the whole interval 0 to T.

Try itCapital R vs lowercase r

One slice of time vs. one rate for the whole stretch

Capital R · one slice eachfuture one-year rates (not observed today)

R_{1}

yr 0→1

R_{2}

yr 1→2

R_{3}

yr 2→3

R_{4}

yr 3→4

R_{5}

yr 4→5

compressed into

Lowercase r · whole intervalone annualized rate, inferred from today's price

r_{0, T}

yr 0 → T · spans the whole timeline

Capital

R_{t}

$R_{t}$ is the one-year rate that applies to the single year from t−1 to t. Each $R_{t}$ is its own slice.

Lowercase r_{0,T}

$r_{0, T}$ is one annualized rate for the whole interval from today (0) to maturity (T). It collapses all the slices into a single number.

We do not observe the future $R_{t}$ 's today. We observe prices, and infer $r_{0, T}$ .

1.7Section 7

Why $r_{0, T}$ is a geometric average of future one-year rates

If we could see the future sequence of one-year rates, a T-year pure discount bond could be priced by discounting through each one-year rate in the chain. But we cannot see the future. Instead, we observe today's bond price and face value. So we define today's T-year spot rate as the single annualized rate that gives the same price. It is terminology plus an accounting identity — but the identity is powerful.

If future one-year rates were known

P_{0} = \frac{F}{( 1 + R _{1} ) ( 1 + R _{2} ) \dots ( 1 + R _{T} )}

Discount the face value through each future one-year rate, one after another.

We cannot observe R₁, R₂, ..., $R_{T}$ today. We only observe today's bond price.

What we actually observe and define

P_{0} = \frac{F}{( 1 + r _{0, T} ) ^{T}}

$r_{0, T}$ is the single annualized rate that gives the same price as the full chain of one-year rates.

$P_{0}$: price today (observed in the market)
$F$: face value paid at maturity
$r_{0, T}$: today's T-year spot rate (backed out from price)
$T$: time to maturity in years

$r_{0, T}$ is a geometric average, not a simple arithmetic average. The observable little r contains information about the market's view of the future path of rates.

Try itChain → single rate

A spot rate is a chain of one-year rates, collapsed

Chain of one-year rates

R₁

R₂

R₃

R₄

R₅

Each R is one slice. Multiplied together, they compound across the whole timeline.

If future one-year rates were known

P_{0} = \frac{F}{( 1 + R _{1} ) ( 1 + R _{2} ) \dots ( 1 + R _{T} )}

If future one-year rates were known.

We observe today's price

P_{0} = \frac{F}{( 1 + r _{0, T} ) ^{T}}

We observe today's price, so define $r_{0, T}$ .

r_{0,T} is a geometric average of one-year rates. Compounding once at r for T years must equal compounding once at each R_t for its own year.

1.8Section 8

STRIPS spot-rate extraction

STRIPS behave like pure discount bonds because each one has no intermediate coupon payments. That makes them useful for extracting spot rates. Here is real MIT data from 2001-08-01. Click a row, and watch the spot rate emerge from the price.

Definition · STRIPS spot rate extraction

A STRIPS discount bond pays exactly $1 at maturity. From its price

P_{0}

and maturity

T

, we solve for today's spot rate

r_{0, T}

Zero-coupon bond price

P_{0} = \frac{F}{( 1 + r _{0, T} ) ^{T}}

Solve for the spot rate (F = 1)

r_{0, T} = (\frac{1}{P _{0}})^{1/ T} - 1

$P_{0}$: price today (per $1 face)
$T$: maturity in years
$r_{0, T}$: today's T-year spot rate

Try itSTRIPS prices · 2001-08-01

Pick a maturity, extract its spot rate

Maturity	Price / $1	Spot rate
	0.991	3.68%
	0.983	3.49%
	0.967	3.41%
	0.927	3.86%
	0.797	4.64%
	0.605	5.15%
	0.187	5.75%

Worked substitution · 5-Yr

0.797 = \frac{1}{( 1 + r ) ^{5}}

r = (\frac{1}{0.797})^{1/5} - 1

r_{0, 5} \approx 4.64%

$1 future payment travels back through the discount tunnel

Today

$0.797

longer maturity → longer tunnel → more discounting

Year 5

$1.000

The $1 payment is fixed; today it is worth 79.7% of face → spot rate 4.64%

Spot rate curve (selected maturity highlighted)

Worked example · 5-year STRIPS (MIT 15.401)

The 5-year STRIPS costs 0.797 per $1 of face. With $F = 1$ : $0.797 = \frac{1}{( 1 + r ) ^{5}}$ , so $r = (\frac{1}{0.797})^{1/5} - 1 \approx 4.64%$ .

1.9Section 9

From many spot rates to the term structure

If we observe prices for many discount bonds today, we can infer many spot rates: $r_{0, 1}$ , $r_{0, 2}$ , $r_{0, 5}$ , $r_{0, 10}$ , and so on. This mapping of maturity to rate is the term structure of interest rates. When plotted, it is a yield curve.

An upward-sloping curve suggests that longer maturities have higher average rates. This may reflect expected future rate increases, inflation expectations, or compensation for lending longer. A downward-sloping curve suggests lower future rates or strong demand for long-term bonds — but it does not guarantee the future.

Definition · Term structure of interest rates

Observing many discount bond prices (

P_{0, 1} \dots P_{0, T}

) lets us infer the set of spot rates (

r_{0, 1} \dots r_{0, T}

) — this maturity-to-rate mapping is the term structure.

Maturity → rateYield curveNot a guarantee of the future

A yield curve plots rates against maturities. An upward curve often suggests higher future rates and/or term premia; a downward/inverted curve suggests lower future rates and/or risk and liquidity effects. But the curve does not guarantee the future.

Try itPrices → rates → curve

Transform prices into rates

Discount prices (input)

1 yr0.9615
2 yr0.9246
3 yr0.8830
5 yr0.7835
7 yr0.6950
10 yr0.5850
20 yr0.3500
30 yr0.2150

Calc engine

r = (\frac{1}{P})^{1/ T} - 1

1 yr→4.00%

2 yr→4.00%

3 yr→4.23%

5 yr→5.00%

7 yr→5.34%

10 yr→5.51%

20 yr→5.39%

30 yr→5.26%

Each price becomes a spot rate. Plotting rate vs maturity gives the term structure.

Upward curve

Curve shape

Simple read

Rates are expected to rise.

Careful read

May reflect expected future rates plus compensation for maturity/liquidity risk.

1.10Section 10

Crisis-era yield curve interpretation

The lecture discusses a crisis period where very short Treasury rates became extremely low because investors rushed into safe, liquid Treasury bills. The professor's point is sharp: a very low T-bill yield does not necessarily mean the world is calm. It can mean everyone is trying to buy the safest short-term instrument at the same time — and that rush drives prices up and yields down.

Definition · Flight to liquidity

When fear rises, investors pile into the safest, most liquid short-term instruments — like T-bills. That buying pushes T-bill prices up and their yields down, even when nothing about the government's credit changed.

Try itStress the short end

The same curve bends differently under different stresses

1m T-bill yield

2.00%

2y yield

3.90%

10y yield

5.20%

A very low T-bill yield can mean everyone is buying the safest short-term instrument at once — not that the government suddenly became more creditworthy.

02Chapter 2

Forward rates and hedging

Spot rates price money from today to a future date. Forward rates price money between two future dates. Once we can compute forward rates, we can lock in future borrowing or lending — and the CFO example shows why a non-speculator would want to do exactly that.

2.1Section 11

Why forward rates appear

Definition · Forward interest rate

Forward interest rates are today's rates for transactions between two future dates. A forward transaction is agreed today, begins at a future date, and ends at a later future date.

Before we use the symbol $f_{t}$ , decode it: a forward rate is agreed today but applies between future dates. The one-year forward rate $f_{2}$ is the rate agreed today for lending or borrowing from year 1 to year 2. Future spot rates can be different from today's corresponding forward rates. The forward rate is known today; the future spot rate is realized later.

Definition · Forward rate

Forward rates are today's rates for transactions between two future dates. You agree on the rate now; the money moves later.

Try itSpot vs. forward

Spot starts now. Forward starts later.

Spot rate

A spot rate governs money that moves now. You hand over cash today; you receive a known amount at a future date.

r_{0, T}

Forward rate

A forward rate $f_{t}$ governs money that moves later. The rate is fixed today, but both the loan and repayment happen between two future dates.

f_{t_{1}, t_{2}}

Heads up: future spot rates can differ from today's forward rates. A forward rate is what you can lock in now — not a guarantee of the rate that will actually prevail then.

2.2Section 12

Forward rate from spot rates: two paths

If we know the 1-year spot rate and the 2-year spot rate, we can infer the one-year forward rate from year 1 to year 2. The logic is no-arbitrage: if both paths to year 2 are locked today and have the same risk, they should lead to the same terminal value.

Path A: Invest for two years at $r_{0, 2}$ .

Path B: Invest for one year at $r_{0, 1}$ , then lock today the forward rate $f_{2}$ from year 1 to year 2.

Forward rate from spot rates

1 + f_{2} = \frac{( 1 + r _{0, 2} ) ^{2}}{1 + r _{0, 1}}

No-arbitrage: both paths to year 2 must give the same result.

f_{2} = \frac{1.0 7 ^{2}}{1.05} - 1

f₂ ≈ 9.04%

The two-year rate is 7% per year over two years. If year 1 is 5%, the implied second-year forward rate must be higher than 7% so the two-year average works.

Try itBuild the forward rate

Two ways to reach Year 2 must agree

1-year spot rate (r₀,₁)5.0%

2-year spot rate (r₀,₂)7.0%

Path A · one move

2 years at

r_{0, 2}

today$1.00

year 2$1.1449

(1 + 7.00%)²

Path B · two moves

1 year at

r_{0, 1}

, then forward f₂

today$1.00

year 1$1.0500

year 2$1.1449

(1 + 5.00%)(1 + 9.04%)

Implied year-2 forward rate (f₂)

9.04%

If year 1 is 5.0%, the implied year-2 forward must be above 7.0% — the 2-year rate is an average, and the first year is the cheap one.

No-arbitrage forward

1 + f_{2} = \frac{( 1 + r _{0, 2} ) ^{2}}{1 + r _{0, 1}}

f_{2} = \frac{1.070 0 ^{2}}{1.0500} - 1

f₂ ≈ 9.04%

If year 1 is 5%, the implied year-2 forward must be above 7%.

2.3Section 13

The CFO hedge example

You are CFO of a U.S. multinational. You expect to receive $10 million from a foreign subsidiary in one year. You will use that money to pay dividends one year after that. You do not know what interest rates will be in one year, but you want to lock in the lending rate from year 1 to year 2.

Current rates: $r_{0, 1} = 5%$ and $r_{0, 2} = 7%$ . Here is the key question: is 7% the rate you care about?

No. 7% is today's two-year spot rate. You care about the one-year rate from year 1 to year 2. That future spot rate is unknown today. But the forward rate $f_{2} \approx 9.04%$ can be locked today.

This example is intentionally confusing until you draw the timeline. Finance is not a spectator sport; you have to track where the money comes from and where it goes. And yes — the hard part is first having the $10 million.

Definition · Hedging with forward rates

By combining a short-term borrow and a long-term lend, a CFO can lock in a one-year forward lending rate for a future period — without knowing what rates will actually be.

Try itCFO finance desk

1Problem2Borrow3Invest4Locked

Lock the Year 1 → Year 2 lending rate

You are the CFO. A foreign subsidiary will repatriate $10MM in Year 1, which you plan to pay as dividends in Year 2. You want to lock in a one-year lending rate from Year 1 to Year 2 — but the Year 1 rate is uncertain today. Current rates: r(0,1) = 5%, r(0,2) = 7%.

Cash flow ($MM)	Year 0	Year 1	Year 2
1-Yr Borrowing @5%	0	0	0
2-Yr Lending @7%	0	0	0
Repatriation	0	+10.000	0

Uncertainty cloud

The Year 1 spot rate is unknown today. If it falls, you lend at a lower rate; if it rises, you lend higher. Start by borrowing today.

Core lesson: Hedging removes uncertainty. It does not guarantee regret-free outcomes. As CFO, your job is usually not to speculate on rates. The point is to solve the financing problem.

2.4Section 14

The bank forward loan example

A customer wants a forward contract to borrow $20 million three years from now for one year. You are the bank. Quote the forward loan rate. The tool is synthetic replication: use discount bonds you can trade to create the cash flows you need.

Forward rate from year 3 to year 4

f_{4} = \frac{( 1 + r _{0, 4} ) ^{4}}{( 1 + r _{0, 3} ) ^{3}} - 1

The forward rate for the period from year 3 to year 4, derived from today's spot rates.

f_{4} = \frac{1.0 7 ^{4}}{1.06 5 ^{3}} - 1

f₄ ≈ 8.51%

Buy 3-year bonds (receive $20MM in year 3), finance by short-selling 4-year bonds (pay $21.7MM in year 4). The implied rate on the synthetic forward loan is 8.51%.

Definition · Synthetic forward contract

A bank can create a forward loan synthetically by buying a shorter discount bond and short-selling a longer one. The cash flows at Year 0 cancel, leaving money received in one future year and repaid in a later one — a forward loan at the forward rate.

Short selling

Short selling means selling a security you do not own by borrowing it or creating the obligation. You receive proceeds now but must deliver the promised future payoff later.

Maturity t (yr)	Price P_t	Spot r(0,t)
1	0.9524	5.00%
2	0.8900	6.00%
3	0.8278	6.50%
4	0.7629	7.00%

Forward rate f₄ (Year 3 → Year 4)

f_{4} = \frac{( 1 + r _{0, 4} ) ^{4}}{( 1 + r _{0, 3} ) ^{3}} - 1

f_{4} = \frac{1.0 7 ^{4}}{1.06 5 ^{3}} - 1 \approx 8.51%

f₄ ≈ 8.51%

Try itBuild the synthetic forward loan · step by step

Quote the Year 3 → Year 4 loan

A customer wants a forward contract to borrow $20MM three years from now, repaying one year later. You are the bank. Build the loan synthetically and quote the rate.

Step 1 of 7

Step 1 — Which bond delivers $20MM in Year 3?

The customer needs $20MM in Year 3. A discount bond that matures in Year 3 pays its face value then. So buy a 3-year discount bond with $20MM face.

3-year discount bond, face = $20MM

Bond blocks on the timeline

03Chapter 3

Coupon bonds and yield-to-maturity

Now we move from pure discount bonds to coupon bonds. Coupon bonds make intermediate payments and repay principal at maturity. They are really packages of dated cash flows — and each dated cash flow should be discounted with the appropriate rate for that date. Practitioners often summarize all of that with one number: yield-to-maturity.

3.1Section 15

Coupon bonds are packages of dated cash flows

A 3-year bond with $1,000 face value and a 5% coupon rate has cash flows of $50 in year 1, $50 in year 2, and $1,050 in year 3. But think about it differently: this bond is really three separate claims — $50 at year 1, $50 at year 2, and $1,050 at year 3. Each claim is a mini zero-coupon bond.

Definition · A coupon bond is a package

A coupon bond is not one instrument — it is a package of dated cash flows. Each payment is, on its own, a tiny zero-coupon bond. STRIPS literally separate them.

Try it3yr · 5% coupon · $1,000 face

One bond, three dated claims

Whole coupon bond

3yr · 5% · $1,000

three promised payments bundled together

Bundled together, the three payments trade as a single coupon bond. The bond's price is just the sum of the prices of its three pieces.

3.2Section 16

Yield-to-maturity is a convenient summary, not the full story

The theoretically clean method discounts each cash flow using the appropriate spot rate for that date. But practitioners often quote coupon bonds using one number: $y$ , the yield-to-maturity. It is a complex average of future spot rates — a summary, not the full term structure.

Solving for YTM is not as simple as solving a zero-coupon bond. It can become a polynomial problem. For normal bonds with positive price and positive cash flows, a meaningful yield generally exists — but this is still a numerical calculation, not a closed-form formula.

Yield-to-maturity

P_{0} = k = 1 \sum T \frac{C _{k}}{( 1 + y ) ^{k}}

The single discount rate y that makes the present value of all coupon and principal payments equal the bond's market price.

$P_{0}$: current market price
$C_{k}$: cash flow at time k
$y$: yield-to-maturity
$T$: maturity

YTM is a complex average of future spot rates. For pure discount bonds, YTM equals the current spot rate. Bond prices and yields move in opposite directions.

Definition · Yield-to-maturity (YTM)

The single discount rate that makes the present value of all promised coupon and principal payments equal the bond's market price.

Yield-to-maturity

P_{0} = k = 1 \sum T \frac{C _{k}}{( 1 + y ) ^{k}}

The single discount rate that makes PV of all cash flows equal the market price.

$P_{0}$: market price
$C_{k}$: cash flow at time k
$y$: yield-to-maturity
$T$: maturity

What YTM is

• A complex average of future spot rates.
• No closed form for coupon bonds — solved numerically.
• Given P₀ + cash flows → solve y. Given y + cash flows → solve P₀.
• For a pure discount bond, YTM = the current spot rate.

Premium / par / discount

• coupon > YTM → premium (price > par)
• coupon = YTM → par
• coupon < YTM → discount (price < par)

Try itYTM solver

Bond prices and yields move in opposite directions

Face value$1,000.00

Coupon rate5.00%

Maturity (years)3 yr

Frequency1/yr

Market price$1,000.00

Solved YTM

5.00%

Price vs par ($1,000.00)

$1,000.00

par

Price–yield curve · price and yield move oppositely

Period	1	2	3
Cash flow	$50.00	$50.00	$1050.00

Drag the price down: the YTM rises, and the badge flips to discount. Drag the price up: the YTM falls and the badge flips to premium. YTM is the single rate that reprices all of these cash flows at the market price.

3.3Section 17

Yield curves from coupon bonds are useful proxies

Public yield curves often use coupon-bearing Treasury securities, not pure STRIPS. That means the plotted yields are YTMs, not pure spot rates. This is a reasonable proxy when coupons are not too different — but strictly speaking, it is not the same as a zero-coupon spot curve.

Definition · What the published curve really is

The Treasury's daily yield curve is built from coupon-bearing bonds, not pure STRIPS. The plotted yields are yields-to-maturity — a reasonable proxy for spot rates, but not identical.

Try itSpot vs. YTM curves

STRIPS spots sit above coupon-bond YTMs

STRIPS spot curve (true zeros)

Coupon-bond YTM curve (published)

Public yield curves use coupon Treasuries, not pure STRIPS. The plotted yields are YTMs, not pure spot rates — a reasonable proxy but not identical. Because each coupon gets discounted at one blended rate, the YTM curve smooths over the true term structure.

04Chapter 4

Yield-curve theories and coupon bonds as STRIPS portfolios

4.1Section 18

Models of the term structure

There are models that try to explain why the yield curve slopes upward, downward, or bends. None fully explains everything. If someone really has a model that predicts yield-curve movements well, that model is valuable — in financial firms, such models may become trade secrets rather than published formulas.

Expectations Hypothesis

$E_{0} [R_{k}] = f_{k}$ — Today's forward rate is the best forecast of the future spot rate.

Liquidity Preference

$E [R_{k}] = f_{k} - Liquidity Premium$ — Long-term borrowing requires a premium because investors prefer liquidity.

Definition · Expectations Hypothesis

E_{0} [R_{k}] = f_{k}

— forward rates are unbiased forecasts of future spot rates.

Definition · Liquidity Preference

E [R_{k}] < f_{k}

f_{k} - Liquidity Premium

. Lenders demand compensation for giving up liquidity.

Try itMatch each scenario to a theory

Term structure theory arena

“Forward rates are the market's best estimate of future spot rates.”

“Long-term borrowers must pay extra because investors prefer liquidity.”

“Some investors prefer specific maturity ranges.”

“Different maturity markets are partly separated by the rules investors must follow.”

“A mathematical model describes rate dynamics over continuous time.”

4.2Section 19

Coupon bonds as portfolios of pure discount bonds

Theorem: All coupon bonds are portfolios of pure discount bonds. A 3-year 5% bond with $1,000 face value is equivalent to 50 one-year STRIPS, 50 two-year STRIPS, and 1050 three-year STRIPS. Each STRIP pays $1 at its maturity — so the portfolio exactly reproduces the coupon bond's cash flows.

Definition · Portfolio theorem

All coupon bonds are portfolios of pure discount bonds. Each coupon is just a zero-coupon bond that pays $1 at its maturity.

Try itDecompose a 3-yr 5% bond

A coupon bond is a bundle of STRIPS

Coupon bond (face $1,000, 5%, 3 yr)

Replicate activity — match the coupon bond with STRIPS

How many $1-face STRIPS of each maturity reproduce the cash flows $50 /, 50 /, 1050$ ?

1-yr STRIPS0

target $50

2-yr STRIPS0

target $50

3-yr STRIPS0

target $1050

4.3Section 20

Transition to arbitrage

If the coupon bond and the matching STRIPS portfolio produce identical future cash flows, their prices should be the same. If they are not, there may be arbitrage. This sets up Lesson 3.3, which covers the law of one price and interest-rate risk in full.

Definition · Law of one price (preview)

Identical future cash flows should have identical prices. A coupon bond and its matching STRIPS portfolio deliver the exact same dollars on the exact same dates — so they should cost the same.

Try itArbitrage preview

If two identical cash-flow packages disagree on price

Coupon bond price

STRIPS portfolio price

Correct

The two packages are priced identically. No trade — the market is internally consistent here.

Idealized. Real markets have frictions — bid-ask spreads, transaction costs, financing, and shorting constraints — that can keep small gaps from being truly free money. Lesson 3.3 covers the law of one price fully.

05Mastery

Summary and mastery check

Try itLesson 3.2 mastery check

Pass with 6 of 8 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
A 5-year zero-coupon bond trades at 0.797 per $1 face value. What information does that price contain?
02
What does $r_{0, 5}$ mean?
03
What does capital R₂ represent?
04
Why is $r_{0, T}$ described as a geometric average of one-year rates, not an arithmetic average?
05
What is the key difference between a forward rate and a future spot rate?
06
In the CFO hedge example, why is 7% (today's 2-year spot rate) NOT the rate the CFO cares about?
07
What is yield-to-maturity (YTM)?
08
A 3-year 5% coupon bond with $1,000 face can be replicated by what STRIPS portfolio?

Lesson summary

1A zero-coupon bond price tells us how the market values one future dollar at one future date.
2Today's T-year spot rate is backed out from a T-year zero price — it is a geometric average of future one-year rates.
3Capital R means a one-year rate for one period. Lowercase r means a multi-year rate observed today.
4The term structure maps maturities to rates. Plotted, it is a yield curve — informative, but not a crystal ball.
5Forward rates are today's rates for future transactions. They are not guaranteed future spot rates.
6YTM is a convenient single-number summary of a coupon bond, not the full spot-rate curve.
7Coupon bonds can be decomposed into zero-coupon bonds. Identical cash flows should have identical prices — otherwise arbitrage.

Back to Bond Markets and Discount Bonds Replay this lesson Continue to Arbitrage, Duration, and Convexity

Fixed-Income Securities

Prices, spot rates, and the term structure

Market prices are thermometers, not crystal balls

From scattered prices to a structured curve

Start from the old problem: a pure discount bond

One payment, one date, one rate

Why different horizons need different rates

Buy a dollar in the future. Read the rate off the price.

What is a zero-coupon bond, and what is a 5-year zero?

Translate a price into today's spot rate

Decode spot-rate notation before using it

What do the subscripts mean?

Capital R versus lowercase r

One slice of time vs. one rate for the whole stretch

Why r0,T​ is a geometric average of future one-year rates

A spot rate is a chain of one-year rates, collapsed

STRIPS spot-rate extraction

Pick a maturity, extract its spot rate

From many spot rates to the term structure

Transform prices into rates

Crisis-era yield curve interpretation

The same curve bends differently under different stresses

Forward rates and hedging

Why forward rates appear

Spot starts now. Forward starts later.

Forward rate from spot rates: two paths

Two ways to reach Year 2 must agree

The CFO hedge example

Lock the Year 1 → Year 2 lending rate

The bank forward loan example

Quote the Year 3 → Year 4 loan

Step 1 — Which bond delivers $20MM in Year 3?

Coupon bonds and yield-to-maturity

Coupon bonds are packages of dated cash flows

One bond, three dated claims

Yield-to-maturity is a convenient summary, not the full story

Bond prices and yields move in opposite directions

Yield curves from coupon bonds are useful proxies

STRIPS spots sit above coupon-bond YTMs

Yield-curve theories and coupon bonds as STRIPS portfolios

Models of the term structure

Term structure theory arena

Coupon bonds as portfolios of pure discount bonds

A coupon bond is a bundle of STRIPS

Transition to arbitrage

If two identical cash-flow packages disagree on price

Summary and mastery check

Why $r_{0, T}$ is a geometric average of future one-year rates